Introduction to Polynomials

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SWBAT recognize the volume formula as a Cubic Function and successfully maximize the volume of a box for a given surface area.

Big Idea

Students work with familiar objects, learning to use mathematics to understand them in closer detail.


25 minutes

At the start of this lesson I provide each student with an Introduction to Polynomials worksheet. In this lesson students need to be able to do the following:

  • Find the volume of a box given the dimensions
  • Write an expression for the width, length, and height of the box in terms of x.
  • Write the volume of an open topped box in terms of x.
  • Find the dimensions that provide the maximum possible volume for the box.
  • Explain what happens to the volume when the sides of the box are doubled.

Problem 1 is a warm-up problem. It helps to remind my students how to find the volume of a box by multiplying length times width times height. Once they complete this task I will hand each pair of students an 8.5 X 11 inch piece of Computer Paper and a ruler to start working on Questions 2-4. I want my students to be able to work with a physical model. I also provide each student a Graphing Calculator, and an individual white board with a marker to write down their thoughts until they are ready to record their actual method on paper.

The primary problem solving task asks students to find the maximum volume of an open topped box.  I will have my students work with their table partners for this task. The task is exploratory and my students may use any method that makes sense to them to determine the dimensions that they think maximize the volume of the box.  

Teacher's Note: When I teach this lesson I have not yet taught my students how to multiply polynomial. Some know how, but I expect many of my students will work with a formula written in factored form. I allow my students to use a calculator for this problem., so it is easier for them to take risks if they have an idea that they want to try. I will, however, be on the lookout for students who are engaging in trial-and- use of the calculator. I want them to work with a plan, whether they use a table, a formula, a graph, etc.  


Partner Work Time

15 minutes

This task has a soft launch. Once all students are working, I continually walk around the room to monitor their progress.  At first, I am observing to make sure that all students write correct expressions for the length, width and height of a box using x as a parameter. The dimensions should be the following:

Length: 8.5-2x

Width:  11-2x

Height: x

The volume function (in terms of x) should be equivalent to:

 V(x) = x(8.5-2x)(11-2x)

Once I have questioned students and every group seems to have the right expressions and equation, I will engaging with pairs more directly to question them about their ideas for maximizing the volume of the box. As I learn about their strategies I will begin selecting groups to share out their methods at the end of the lesson. I will select groups that help the class to recognize and appreciate the different methods that can be used.

The final question (#4) is meant to challenge my students to reason. Students are asked to predict the volume of the open box if the measurements of the sides are doubled. If none of my students demonstrate a method using the graphing calculator, then I will demonstrate how ot use the calculator in a way that is similar to that shown in the video below.

Exit Slip

10 minutes

After the students share out their different methods for finding the dimensions that create the maximum volume, I hand them an Exit slip.  I expect the Exit Slip to take them about 10 minutes to complete.  I will use the Exit Slip as a formative assessment to check for student understanding of volume.  This a two step problem:

  1. Students need to find the volume of one candy bar my multiplying the length by width by height.  
  2. Students need to calculate how many candy bars can fit into 66.15 cubic inches.  

The volume of one candy bar is 2.85546875 cubic inches.  The quotient of 66.15 divided by 2.85546875 is approximately 23.166 candy bars.  This result means that 23 candy bars can fit into the box.  

We discuss the importance of not rounding until the final answer.  Using more decimal places until the final answer makes our conclusion more accurate.